In an infinite dimensional semi-flow setting the number \(n(u_*)\)  is usually finite (although the number of positive eigenvalues is infinite now), and one may again use \(i(u_*)=n(u_*)\)  as the index, and the resulting construction is usually still called a Morse homology.
In other cases, so-called strongly indefinite problems, however, the linear operator associated to the second derivative of \(\mathcal {A}\)  has both infinitely many negative and infinitely many positive eigenvalues, hence an alternative definition of the index is needed. The breakthrough idea put forward by Floer [1]}, [2]} in
the context of Hamiltonian dynamics, is to define a relative index: we have no absolute index but only ever compare the indices of two critical points, say \(u_*^1\)  and \(u_*^2\) . In essence this relative index \(i(u_*^1,u_*^2)\)  counts how many eigenvalues cross over from negative to positive
when one follows a path of linear operators along a homotopy from \(d^2 \!\mathcal {A}(u_*^1)\)  to \(d^2 \! \mathcal {A}(u_*^2)\)  — spectral flow (e.g. see [3]}). Once such a relative index is properly defined (e.g. taking into account eigenvalues can cross 0 in both directions), the homology construction can be carried out under suitable compactness and generic transversality conditions. For strongly indefinite problems the resulting homology, based on a relative index rather than a Morse index, is usually called a Floer homology. Clearly Floer homology is a generalization of Morse homology, whereas the Morse index can be seen as a special case of a relative index: \(i(u_*^1,u_*^2) = n(u_*^1)-n(u_*^2)\) . On the other hand, by choosing a fixed critical point \(u_*^0\) , one can attach an index to any critical point by setting \(\tilde{n}(u_*)=i(u_*^0,u_*)\) ,
and one may even choose any fixed hyperbolic linear operator as the “base point” rather than an equilibrium, see Section  for more details. We will refer to the collective of such constructions as Morse-Floer theory.
